TITLE: About the sum of rectangular power sums
QUESTION [7 upvotes]: Let $n \geq 1$ be an integer and consider the symmetric function
$$D_n = \sum_{d|n} p_d^{n/d},$$
where $p_{d}$ are the power-sum symmetric functions.
It can be checked up to $n=35$ that the symmetric function $D_n$ is Schur-positive.
The multiplicity of the Schur function $s_n$ in $D_n$ is given by the number of divisors of $n$ (sequence A000005 in the OEIS).
One can also easily compute the multiplicity of the Schur function $s_{1^n}$ in $D_n$:
$$\sum_{d|n} (-1)^{n+d},$$
which is sequence A112329 in the OEIS.

Q1: Is the Schur-positivity of $D_n$ for all $n$ known ?


Q2: Has this been considered somewhere in the literature ?

I have alas not much more context for this.

REPLY [4 votes]: Just adding that the expression for the Frobenius characteristic of $\theta_d$ is classically attributed to H. O. Foulkes, see e.g. Richard Stanley's EC2, Problem 7.88.